Math Table

Putnam Post Mortem

Speaker: Noam Elkies

Abstract: Join our own Professor Noam Elkies to discuss a selection of this year’s Putnam problems.

For food, please RSVP here.

Nontransitive and Balanced Sets of Dice

Speaker: Josh Rooney

Abstract: Let's recall the game "rock-paper-scissors." Although rock beats scissors and scissors beats paper, rock does not beat paper. We call this property "nontransitivity." Does this same idea translate to dice? In 1970, Martin Gardner popularized a set of 4 dice (A,B,C,D) "Efron dice" in which P(B>A)= P(C>B) = P(D>C) = P(A>D) = 2/3. Since these probabilities are greater than 1/2, this set of dice is nontransitive, and since the probabilities are equal, we will call the set "balanced." The dominance probability, 2/3 in this case, that is possible for such a set of dice is dependent on the number of dice in the set. In this talk, we will fully classify all such probabilities for a set of n nontransitive and balanced dice. This talk will aim to be accessible to all undergraduates.

Universal Constructions: Drinking the Kool-Aid of the Universe

Speaker: Peter Chon

Abstract: Mathematics is full of seemingly unrelated structures which can be unified via universal constructions. In this talk, we’ll dive into the concept of universality, revealing how universal properties—often seen as isolated tricks in contexts like kernels and free objects—provide deep connections across fields. By exploring examples accessible to all undergraduates, we will show how universal constructions simplify relationships, clarify definitions, and connect diverse areas of math. We will introduce fundamental category theoretic notions from the ground up. Come experience the transformative power and elegance that universal properties bring to mathematics. Once you’ve tasted this “Kool-Aid,” you may never see math the same way again.

Representation theory via geometry

Speaker: Ari Krishna

Abstract: Given a group G, a fundamental question is how to construct and understand its irreducible representations. A rich class of finite simple groups for which we can ask this question is the finite groups of Lie type, close cousins of algebraic groups over finite fields. In this talk, we will focus on the group SL_2(F_q) – through the process of parabolic induction, we obtain roughly half of its irreducible representations, so how can we excavate the others? Drinfeld realized that the answer to this lay in the geometry of an “an extremely symmetric” algebraic curve over F_q, a subtle and beautiful idea that inspired Deligne and Lusztig to prove far-reaching results about representations of all finite groups of Lie type by analyzing algebraic varieties that now bear their names. We retrace the roots of Deligne-Lusztig theory by studying SL_2(F_q), and if time permits, discuss more general developments.

Connectivity properties of Hamiltonian Graphs

Speaker: Quinn Brussel

Abstract: Given a graph G, the “Frank number” F(G) encodes interesting connectivity properties of the graph. It is conjectured that for all cubic graphs, the Frank number is at most three. In the service of this main conjecture is the conjecture that if G is Hamiltonian and cubic, then F(G) is at most 2. In this talk we will show that this result is true.

Which simple shapes wear simple sweaters?

Speaker: Eric Shen

Abstract: In this expository talk I'll discuss some results on which "simple" manifolds, e.g. lens spaces, are the boundaries of other "simple" manifolds. Along the way we'll learn about the intersection form, Betti numbers, Dehn surgery, and what lens spaces actually are.

On graphs that are tough but not prism-hamiltonian

Speaker: Milligan Grinstead

Abstract: Determining if a graph is Hamiltonian is a long-standing problem, dating as far back as chess-related games in the 9th century. A common approach is to ask if we can impose certain constraints such that the Hamiltonian problem becomes solvable. The research presented in this talk was inspired by placing constraints on a condition called Toughness. This talk will introduce Toughness as well as a few other conditions and discuss their implications on the Prism-Hamiltonicity of a graph, a broader version of the Hamiltonian problem. During the talk we will also answer an open question in the field and present a few of our own.

The 3-body problem

Speaker: Laura DeMarco

Abstract: The field of dynamical systems has a long and fascinating history, originating with the study of planetary motion.  It has become a central part of mathematics today, with many connections to algebra, geometry, and analysis.  I will present some of its historical development, with emphasis on the subtle question of linearization and how that leads to deep and difficult problems that remain unsolved today.